Mechanical valve tappets of the above generic type have been known for a long time. Thus, for example, DE-AS 22 47 069 shows a cup tappet which, on the surface of its bottom facing away from the cam comprises an elevation pointing towards a valve stem and bearing with the help of a surrounding shim against the end of the valve stem. The required correction of valve clearance is effected after a defined operational performance by replacing the shim. A drawback of the said structure is that, on the one hand, the shims have to be manufactured and stocked in different thicknesses as additional parts and, on the other hand, the adjustment of valve clearance is made more difficult because the shim is a separate part necessitating additional handling.